Let p: ‘If x is a real number such that x^{3} + 4x = 0, then x is 0’

q: x is a real number such that x^{3} + 4x = 0

r: x is 0

**(i) **We assume that q is true to show that statement p is true and then show that r is true

Therefore, let statement q be true

Hence, x^{3} + 4x = 0

x (x^{2} + 4) = 0

x = 0 or x^{2} + 4 = 0

Since x is real, it is 0.

So, statement r is true.

Hence, the given statement is true.

**(ii) **By contradiction, to show statement p to be true, we assume that p is not true.

Let x be a real number such that x^{3} + 4x = 0 and let x ≠ 0

Hence, x^{3} + 4x = 0

x (x^{2} + 4) = 0

x = 0 or x^{2} + 4 = 0

x = 0 or x^{2} = -4

However x is real. Hence, x = 0, which is a contradiction since we have assumed that x ≠ 0

Therefore, the given statement p is true.

**(iii)** By contrapositive method, to prove statement p to be true, we assume that r is false and prove that q must be false

∼r: x ≠ 0

Clearly, it can be seen that

(x^{2} + 4) will always be positive

x ≠ 0 implies that the product of any positive real number with x is not zero.

Now, consider the product of x with (x^{2} + 4)

∴ x (x^{2} + 4) ≠ 0

x^{3} + 4x ≠ 0

This shows that statement q is not true.

Hence, proved that

∼r ⇒ ∼q

Hence, the given statement p is true.

Answered by Abhisek | 1 year agoDetermine whether the argument used to check the validity of the following statement is correct: p: “If x^{2} is irrational, then x is rational.” The statement is true because the number x^{2} = π^{2} is irrational, therefore x = π is irrational.

Which of the following statements are true and which are false? In each case give a valid reason for saying so

**(i)** p: Each radius of a circle is a chord of the circle.

**(ii) **q: The centre of a circle bisect each chord of the circle.

**(iii)** r: Circle is a particular case of an ellipse.

**(iv)** s: If x and y are integers such that x > y, then – x < – y.

**(v)** t: \( \sqrt{11}\) is a rational number.

By giving a counter example, show that the following statement is not true. p: “If all the angles of a triangle are equal, then the triangle is an obtuse angled triangle.”

Show that the following statement is true “The integer n is even if and only if n^{2} is even”

Show that the following statement is true by the method of the contrapositive p: “If x is an integer and x^{2} is odd, then x is also odd.”